School of Mathematics and Statistics

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2013/2014 Sem. 1

Aims

This module introduces the ideas and techniques in the fi eld of dimension theory applied to chaotic dynamical systems. We will see how to compute the fractal dimension of dynamically de ned objects, many of which have an extremely rich `multifractal' structure. The signature dimensional behaviour of such systems is related to the degree of turbulence and intermittency the systems exhibit. The mathematical tools here will come from thermodynamic formalism, which has a parallel with a formalism used in statistical physics. Specifi cally, thermodynamic formalism gives us invariant measures which live on dynamically de ned fractal sets. We will study the dimension of these sets through these measures. Topics include a review of Hausdorff dimension, and introductions to entropy, topological pressure and Bowen's formula. The canonical application is to the class of simple interval maps known as `cookie cutters'.  Dimension theory and dynamical systems are active research areas in the school.

Objectives

- To understand, and in standard cases be able to compute, the Hausdorff dimension of probability measures.
- To apply the ideas of thermodynamic formalism, for example pressure, to standard dynamical systems.
- To understand Gibbs measures for uniformly expanding interval maps.
- To understand Bowen's formula: what it means in applications and how it can be proved.

Syllabus

- Hausdorff measure and dimension
- Dynamically de ned Cantor sets: the cookie cutter
- the Mass distribution principle
- ergodic invariant measures
- entropy and topological pressure
- Gibbs measures and their Hausdor ff dimension
- Bowen's formula

Assessment

100% by two-and-a-half hour examination

Textbooks

Equilibrium States and the Ergodic Theory of Anosov Di eomorphisms: Rufus Bowen, Lect. Notes in Math., vol. 470. Springer;
Fractal Geometry: Kenneth Falconer, Wiley;
Equilibrium States in Ergodic Theory: Gerhard Keller;
Conformal fractals: ergodic theory methods: Feliks Przytycki and Mariusz Urbanski, Cambridge University Press;
Dimension Theory in Dynamical Systems: Contemporary Views and Applications: Yakov Pesin, Chicago University Press;
An introduction to Ergodic Theory: Peter Walters, Springer.

MT4004 or MT4515

Availability

Academic year 2012/13 in semester 2 at 10

Prof L Olsen